But all is not lost as far as partition refinement is concerned. We can
generalize the forward algorithm of [1]} and
[2]} to a construction of the final coalgebra in
\(\mathsf {SMod}\)  based on a method of of Adámek and
Koubek [3]} that uses the notion of the final
sequence (or terminal sequence). In general the construction takes
steps indexed by ordinals that can go beyond \(\omega \) , although for semimodules
no more than \(\omega \)  steps are necessary as is shown below, the proof being
identical to that of [2]}. In special cases, such as for finite
dimensional vector spaces, the construction terminates after a finite number of
steps and the required operations can be performed effectively. In this case the
procedure is the same as the forward algorithm of
[1]}, [2]}. We assume from now on that the
set of states \(V\)  in the \(K\) -LWA \((V,\langle o,t \rangle )\)  is an Artinian
semimodule. Our presentation below has a lot in common with that of
[2]}, but also has major differences due to the
less enriched setting of semimodules as compared to vector spaces.
